"In 1977, Grandpa Meyer and I were at Wolfie’s in North Miami Beach, and I noticed two young boys in yarmulkes looking over at us. I was standing behind Grandpa when the boys walked up. One said: ‘Hey, Mr Lansky, we’d like to get your autograph!’ Grandpa paused for a moment. He looked seriously at the boys and said: ‘What did I do? Win an Academy Award?’

“One of the boys looked earnestly at Grandpa and said: ‘Well, we thought it would be worth some money some day.’ Grandpa smiled and replied: ‘Sorry, son, I don’t sign autographs.’

This was how Meyer Lansky II learned that his grandfather was famous. Far from winning an Oscar, Grandpa Meyer was part of the Jewish mafia that ran hand-in-hand with the Italian mob in the US in the first half of the 20th century.

Until he was 12, Lansky II, now 61, had no idea that his grandfather, along with another infamous Jewish gangster, Bugsy Siegel was credited with “creating Las Vegas”. Or that, like “Lucky”Luciano, Al Capone and Vito Genovese, the “Mob’s Accountant” was one of the most notorious organised crime figures of all time.

Lansky II grew up in Washington and one night he was home with his father, Paul watching TV: “Walter Cronkite came on. They showed a map of the countries my grandfather had tried to get into after leaving Israel. I didn’t understand. My dad tried to explain a little bit. It was on TV for quite a few nights. Then finally ‘Meyer Lansky back in Miami Beach, arrested for tax evasion’. I didn’t know what that was. I was saying, ‘what are they doing to Grandpa?’ It was shocking to me. Hearing the name, being his name-sake, and not knowing what was going on.

“I didn’t have a clue. I was scared for him. That was when I started to learn about what he was, his background.”

His grandfather, always a fervent Zionist, had wanted to end his days in Israel. He spent two years there before finally being deported back to the US. Ultimately, he was found not guilty of the charges against him, pretty much like every other occasion when he had been arrested. Meyer Lansky was never found guilty of any crime.

How did young Lansky feel when he discovered the real story of his grandfather, the man who pushed him into getting good grades at school and set mathematical posers at meal times?

“It’s quite something to learn that your grandfather was involved in that kind of life. In the 1970s, I was treated very well, particularly I think because of my grandfather. I remember walking to school and my friends’ brothers were coming out and they were a little older. We’re were all going into Junior high school. You know these kids have their ‘initiation’ things, but they left me alone. If anyone started, they always said, ‘leave him alone.’ I felt a sense of privilege without really understanding why. Until later on, of course.”

Many children and grandchildren of the mobsters have decided to keep a low profile, but that was hard for him to do as he bore his grandfather’s name.

“My dad insisted on calling me Meyer. My grandpa refused when I was born. Of course it is not traditionally Jewish to be called after someone whilst they are still alive, but mainly he didn’t want me to be tainted by the name, he was scared of that. He said ‘I don’t want him to have problems later in life.’ He agreed eventually. He would be surprised to know I think that it’s actually been a great name to have.”

Lansky is upbeat, open and friendly but there may be hidden scars because, for example, he has no photographs of himself with his grandfather apart from as a baby.

“He never wanted to have pictures of me together with him, because he was paranoid that I would have problems especially with the name,” says Lansky, who used to regularly visit his namesake in Miami.

“In my childhood years, clothes were the thing. He would educate me on great clothes; he wore great clothes. He’d show me the proper material, the stitching, he was a very meticulous man. He would ask about my studies, my sports. I have over 400 letters he wrote to my father over a 35-year period. My father gave them all to me and my grandpa asks about me in them all the time.”

He admits there were times he felt apprehensive. “When we would go for dinner we would order things. I’d be a little nervous because he would send things back if they weren’t properly cooked. I always felt a little edgy around him because he was a perfectionist. I was 11 years old, I didn’t want to say anything wrong. I used to follow his lead.”

He also met Vincent “Jimmy Blue Eyes” Alo, who set up casino operations with Meyer Lansky in Florida and Cuba.; “Every time I came to Miami, someone would come and pick me up at the airport, Grandpa never came. I’d get to the house and Jimmy and his friends were all there, all dressed so sharply! They would always be great with me.”

He never met his grandfather’s other ally and business partner Bugsy Siegel, who died in 1947, although his father has talked about him: “Meyer lived on Park West in Miami and Ben (Siegel’s real name) and his wife Phyllis lived on the east side. He would come over when my father was growing up. Seigel didn’t really have time for kids but was always glad to see my dad. ‘Hey, Paul’s here, great,’ he’d say when he came in the door. He didn’t have a New York accent, he’d got rid of that.”

Lansky has seen many actors portray his grandfather or characters based on his grandfather in countless movies about the mob.

“The most iconic film for Grandpa was Godfather II. He not only saw the movie but phoned Lee Strasberg to congratulate and offer light criticism!”

In the film, the character Hyman Roth, played by Strasberg is based on Lansky. “In life, Grandpa spoke with his hands behind his back, and talked baseball but could become irate quickly.” This was all part of Strasberg’s acting, which reminded him strongly of his grandfather. However, he says, “I doubt Grandpa ever saw business guests in his home wearing an unbuttoned shirt. He was a sharp dresser!”

Meyer Lansky was born Maier Suchowljansky in 1902, in Grodno, Russia. His parents, Max and Yetta, named him after a famous rabbi. Following pogroms in Russia, Max decided to move his family to the United States. Max travelled ahead in 1909, and was given the surname “Lansky” on Ellis Island. He quickly set about making a home for his family in Brooklyn. On April 8, 1911, the rest of the family arrived, including Yetta and sons Maier and Jacob.

Throughout his life, although not a religious Jew, Meyer Lansky was proud to be Jewish. When there were Nazi and brown shirt uprisings and gatherings in the 1930s in New York, Lansky organised “associates” to break them up. His grandson says: “People thanked him and he said, ‘You don’t have to thank me, I’m a Jew, why wouldn’t I do this?’”

Lansky donated money to Israel and synagogues; the Jewish Museum of Florida which was once a synagogue is home to a stained-glass window donated by him.

Lansky II decided on a career in the hotel business, working first in Lake Tahoe and then in Las Vegas, but now concentrates solely on promoting his ancestor as a “brand”. He’s actively involved in the Mob Museum in Las Vegas where he lives with his wife Dani, and is bringing out a range of Lansky memorabilia.

“A lot of the children and grandchildren of the others don’t really want to be known for obvious reasons. I was friendly with Millicent Siegel, Bugsy’s youngest daughter. She died last year. Luciano didn’t have any kids that we know of, and he was never married. I know Deirdre Capone, Al Capone’s niece. Frank Costello, who was part of the Luciano ‘family’ was alive until 1973, but I never met him.”

There are many myths and counter myths surrounding Meyer Lansky, who died in 1983 and is buried in a Jewish cemetery in West Miami. Perhaps the most puzzling one is that supposedly he made millions of dollars in his lifetime, once owning hotels and casinos in Las Vegas, but when he died he left only $57,000 in his will. For the first time, in our conversation his grandson is hesitant.

“That was in the will, given to his second wife Teddy. That is what he left on paper,” Meyer Lansky II laughs.